Question: Esther the Clown does face painting at the city carnival. She paints $7$ faces every $21$ minutes and spends the same amount of time painting each face. Write an equation that relates $f$, the number of faces she paints, and $m$, the time she spends painting in minutes.
Explanation: Let's find the constant of proportionality. In the proportional relationship between $f$, the number of faces she paints, and $m$, the time she spends painting in minutes, one constant of proportionality is the number of faces she paints per minute. It is the number we multiply by the time to get the number of faces she paints. $m\,\times\, ?=f$ $\begin{aligned} m\,\times\, {?}&=f \\\\ {?}&=\dfrac{f}{m} \\\\ &=\dfrac{7}{21} \\\\ &={\dfrac{1}{3}} \end{aligned}$ The constant of proportionality is ${\dfrac{1}{3}}$. This means we can multiply ${\dfrac{1}{3}}$ by the time to get the number of faces. Now, let's write the equation: $\begin{aligned} \text{number of faces}&={\text{painting rate}}\times\text{time} \\\\ f&={\dfrac{1}{3}}m \end{aligned}$ One correct equation is: $f = \dfrac 13 m$